Onsager-Machlup functional for stochastic evolution equations

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ONSAGER–MACHLUP FUNCTIONAL FOR STOCHASTIC EVOLUTION EQUATIONS FONCTIONNELLE D’ONSAGER–MACHLUP POUR LES ÉQUATIONS D’ÉVOLUTION STOCHASTIQUES

Xavier BARDINA^a,1, Carles ROVIRA^b,2, Samy TINDEL^c,∗,3

aDepartament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain bFacultat de Matemàtiques, Universitat de Barcelona, Gran Via 585, 08007 Barcelona, Spain cDépartement de mathématiques, institut Galilée – Université Paris 13, avenue J. B. Clément,

93430 Villetaneuse, France

Received 6 May 2000, revised 8 December 2001

ABSTRACT. – We compute the Onsager–Machlup functional for a stochastic evolution equation with additive noise, usingL²-type techniques. The regularity that has to be imposed to the drift coefficient is a trace type condition on its derivative. The expression of the divergence part of the functional does not depend on the stochastic convolution related to our evolution system.

2003 Éditions scientifiques et médicales Elsevier SAS

Keywords: Onsager–Machlup functionals; Stochastic evolution equations

RÉSUMÉ. – Dans cet article, nous calculons la fonctionnelle d’Onsager–Machlup associée à une équation d’évolution stochastique avec bruit additif, en utilisant des techniques de typeL². Les hypothèses de régularité que l’on impose au coefficient de dérive sont des conditions de trace sur sa dérivée. Notons que l’expression de la partie divergence de la fonctionnelle ne dépend pas de la convolution stochastique associée au système d’évolution.

2003 Éditions scientifiques et médicales Elsevier SAS

∗Corresponding author.

E-mail addresses: [email protected] (X. Bardina), [email protected] (C. Rovira), [email protected] (S. Tindel).

1Partially done while the author was visiting the Université Paris VI with a CIRIT grant. Partially supported by grants CICYT PB960088, PB961182, HF1998-0096.

2Partially done while the author was visiting the Université Paris XIII with a SEUID grant. Partially supported by grants CICYT No. PB 960088 and HF1998-0096.

3Partially supported by the grant CIES No. 99071.

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1. Introduction

Let H be a real separable Hilbert space, and consider the following stochastic differential equation onH:

dX(t)= [AX(t)+F (t, X(t))]dt+B dW (t), t∈ [0,1],

X(0)=x, (1)

where x ∈ H, A:D(A)⊂H → H generates a C0-semigroup {exp(tA);t 0}, F is a Lipschitz function defined on [0,1] ×H, B is a non-negative bounded linear operator from H to H, and W is a cylindrical Wiener process inH. It is known that if ₀¹exp(tA)B²H Sdt is finite, then (1) admits a unique solution X∈L²([0,1];H ), in a sense that will be specified later (see [4] for further details). In the remainder of the paper, we will also denote byW^A the stochastic convolution ofAby W, that is the solution to (1) forF ≡0 andx=0.

This article proposes to study the limiting behaviour of ratios of the form γε(φ)≡P (X−φε)

P (W^Aε)

when ε tends to 0, where φ is a deterministic function satisfying some regularity conditions and · is a suitable norm defined on the functions from [0,1] to H. When limε→0γε(φ)=exp(J0(φ))for allφ in a reasonable class of functions, then the functionalJ0is called the Onsager–Machlup functional associated to (1) and · . It is worth noting that J0 can easily be interpreted as a generalized likelihood functional in infinite dimension, which makes its calculation an interesting problem.

For usual stochastic differential equations, namely when H =R^d, A=0, B =Id, the problem of computing the Onsager–Machlup functional for (1) has been widely investigated. Ikeda and Watanabe [9] gave a rigorous proof for the case of any φ ∈ C²([0,1];R^d) for the norm · ∞ defined on C([0,1];R^d). This result has been enhanced then in two directions: Shepp and Zeitouni proved first in [13] that the function φ could be taken in the Cameron–Martin space W₀^1,2([0,1];R^d), and Capitaine proved in [2] and [3] (basing this last result on some techniques inspired by the computation of the Onsager–Machlup functional for diffusions on manifolds, see e.g. [8]) that the norm · could be taken as any euclidian norm on the functions from [0,1] toR^d making sense for the solution to (1) and dominating the norm onL²([0,1];R^d). It is important to note that in the case of finite dimensional diffusions, the functionalJ0does not depend on the norm considered.

Our current work fits in a more global project of studying the Onsager–Machlup functionals for infinite stochastic systems. The first results in that direction have been obtained by Dembo and Zeitouni [5] for trace class elliptic SPDEs on a bounded domain ofR^d, and then by Mayer-Wolf and Zeitouni [11] in the non trace case. We shall use some of their techniques in order to get our main result: if · is chosen as the Hilbert

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norm onL²([0,1];H ), then, forφ satisfying some suitable hypothesis,

J0(φ)= − 1

0

1

2B⁻¹Aφ(t)+Ft, φ(t)− ˙φ(t)²_Hdt−Tr(SP R^∗),

where SP R^∗ is a certain bounded linear operator based on ∇xF and φ. However, in case ∇xF (s, x) is a trace class linear operator for all x ∈H and s ∈ [0,1], then Tr(SP R^∗)=¹₂₀¹[Tr(∇xF )(s, φ(s))]ds.

With respect to the finite dimensional case, some differences can already be stressed in this introduction: first, the generality on the kind of norm considered by Capitaine in [3]

seems beyond our hopes at this moment. On the other hand, since an independence of the functional J0with respect to the norm · can also be expected in our SPDE case, we chose to work with the Hilbert norm onL²([0,1];H )for two main reasons:

1. We are working here with the minimal assumptions on A and B under which Eq. (1) has a unique solution in L²([0,1];H ), though we will also make the additional assumption thatAand B can be diagonalized in the same orthonormal basis ofH.

2. We will be able to use the conditional exponential moments results stated in [11], where the conditioning is over an infinite dimensional Gaussian random variable (these results will be recalled at Section 2). However, the fact that we are dealing with an evolution type equation will force us to delve deeper into the different Karhunen decompositions involved in the application of the results of [11]. We shall give some details about these decompositions at Section 2.

Another relevant difference between the finite and infinite dimensional case is that the normalizing factor inγε(φ)cannot be a function of the norm of the cylindrical Brownian motion. This leads us to the natural choice of a normalization byP (W^Aε), which prives us of the rotational invariance type properties of the Brownian motion used by Ikeda and Watanabe ([9, Lemma 8.2]) and by Capitaine ([2, Lemma 4], [3, Lemma 3]) as a fundamental step towards the computation of the Onsager–Machlup functional.

Our paper is organized as follows: in Section 2, we recall some basic results and fix our notations for the stochastic evolution equation considered. We recall a basic lemma of Mayer-Wolf and Zeitouni [11, Lemma 2.5] that we shall use later on, and give the Karhunen expansion of an Ornstein–Uhlenbeck process in dimension one. In Section 3 we obtain the Onsager–Machlup functional: at Section 3.1 we reduce our problem by Girsanov’s transform and at Section 3.2 we obtain useful decompositions of W^A and Wiener integrals. Section 3.3 is then devoted to some details about the linear case, that is when F is a linear bounded operator, which will lead us to the general case after Taylor’s expansion, and is of independent interest, since the conditions given on F in this case will be more explicit, especially when F can be diagonalized in the same complete orthonormal system thanAandB. At Section 3.4, we will deal with the general nonlinear case.

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2. Notations and preliminary results 2.1. Stochastic evolution systems

2.1.1. The operatorsAandB

Let H be a real separable Hilbert space and A:D(A)⊂H →H an unbounded operator onH. The norm onH will be denoted by| · |H, and the scalar product by·,·. The L² norm in L²([0,1];H )will be denoted by · 2. Let L(H )the set of bounded linear operators onH. The norm · will be the usual operator norm defined onL(H ), that is, T =sup_x_∈_H ^|^{T (x)}_|_x_|^|^H

H .For an operator T ∈L(H ),ST ≡¹₂(T +T^∗)will denote the symmetrized operator based onT. We shall suppose

(H1) The operatorAgenerates a self adjointC0-semigroup exp(tA);t0,

of negative type. Moreover, there exists a complete orthonormal system{ej;j 1}which diagonalizes A. We shall denote by {−αj;j 1}the corresponding set of eigenvalues and we assume that{αj;j 1}is an increasing sequence of real numbers such thatαj0 and limj→∞αj= ∞. Setj0=inf{j, αj >0}. We shall also consider an operatorB∈L(H )satisfying

(H2) The operatorB is of non-negative type and is diagonal when expressed in the orthonormal basis{ej;j1}. We shall denote by{βj;j 1}the corresponding set of eigenvalues. Furthermore, we shall suppose thatβj>0 for allj 1 and that

∞ j=1

β_j² 1+αj

<∞.

In our case where A and B can be diagonalized in the same complete orthonormal system, note that the last hypothesis corresponds to the more general one ₁

0 exp(tA)B²H Sdt <∞, that can be found in [4] in order to ensure the existence and uniqueness of a solution to (1).

2.1.2. Stochastic evolution equations

Let ( ,F,Ft, P ) be a stochastic basis and {W^j(t);t ∈ [0,1], j 1} a sequence of mutually independent Brownian motions adapted to Ft. The cylindrical Brownian motion onH is defined by the formal series

W (t)=^∞

j=1

W^j(t)ej, (2)

where{ej;j1}is the complete orthonormal system ofH introduced at Section 2.1.1.

Note that the series (2) does not converge inH, but for anyh∈H,{W (t), h;t∈ [0,1]}

is a linear Brownian motion with covariance |h|²H (see [4]). We shall consider the stochastic evolution equation (1), whereAandB have been defined at Section 2.1.1 and F:[0,1]×H→His a Lipschitz function with a linear growth condition inxuniformly int (some further hypothesis on F will be made at Section 3). Eq. (1) is only formal,

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and has to be interpreted in the usual mild sense: we will say thatX= {X(t);t∈ [0,1]}

is a solution to (1) if it is anH-valuedFt-adapted square integrable process such that X(t)=exp(tA)x+

t

0

exp(t−s)AFs, X(s)ds+ t

0

exp((t−s)A)B dW (s), (3) for allt∈ [0,1], where the last integral is of Itô’s type. The following proposition is then a particular case of [4, Theorem 7.4].

PROPOSITION 2.1. – Suppose that (H1) and (H2) are satisfied. Then there exists a unique solutionXto (3), such thatX∈L²( × [0,1];H ).

2.2. Some conditioned exponential inequalities

We will recall here some basic lemmae that we shall use for the computation of our Onsager–Machlup functional. The first one is a crucial, though elementary, inequality that can be found in [9, p. 537].

LEMMA 2.2. – For a fixed n1, let z1, . . . , zn be n random variables defined on ( ,F, P )and {Aε;ε >0} a family of sets in F. Suppose that for any c∈R and any i=1, . . . , n, we have

lim sup

ε→∞ Eexp(czi)|Aε

1.

Then

εlim→∞E

exp _n

i=1

zi Aε

=1.

In the sequel of the paper, we shall use some inequalities involving trace class operators. Let us state now the notion of trace that we shall consider.

DEFINITION 2.3. – LetKbe a separable Hilbert space. LetT:K→Kbe a compact symmetric operator. Let{τi;i1}be the eigenvalues of the operatorT. We will say that T is a trace class operator if^∞_i₌₁|τi|is finite.

IfT is trace class, we define the trace of T, Tr(T ), as ^∞_i₌₁T ei, ei for any basis {ei;i1}. In particular Tr(T )=^∞_i₌₁τi.

We finish this subsection recalling two technical lemmas from [11]: the first one is a particular case of [11, Lemma 2.4], and the second one is a slight variation of [11, Lemma 2.5]. We denote here by)²the set of sequences of real numbers{ηi;i1}such that_i₁η_i²<∞.

LEMMA 2.4. – Let{zi;i1} be a sequence of independent N(0,1) random vari- ables defined on ( ,F, P ), and {ηi;i1} and {νi;i1} two )² sequences of real numbers such thatηi=0 for anyi1. Then

εlim→0E

exp _∞

i=1

ziνi ∞ i=1

η²_iz²_i ε

=1.

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LEMMA 2.5. – Let{zi;i1} be a sequence of independent N(0,1) random vari- ables defined on( ,F, P ), and{ηi;i1}a)²sequence of numbers such thatηi=0 for any i1. Let T:)²→)² be a Hilbert–Schmidt operator, {mi;i1} a complete orthonormal system of)², and denoteT mi, mjbyTi,j.

1. IfST ≡¹₂(T +T^∗)is trace class, then

εlim→0E

exp _∞

i,j=1

zizjTi,j ∞

i=1

η²_iz²_i ε

=1.

2. If^∞_i₌₁Ti,i= +∞(respectively−∞), then

εlim→0E

exp

i=j

zizjTi,j+^∞

i=1

(z_i²−1)Ti,i ∞

i=1

η_i²z²_i ε

=0 (respectively+∞).

Proof. – We refer the reader to the proof of [11, Lemma 2.5]. Note only that for any j, i1,

zizj(Ti,j +Tj,i)=zizj

(ST)i,j+(ST)j,i

, where(ST)i,j= STmi, mj. ✷

2.3. The Karhunen–Loève expansion

We compute here the Karhunen–Loève expansion for a class of one-dimensional Ornstein–Uhlenbeck processes that will appear in the decomposition of W^A. The following lemma is presumably fairly standard, but we include it for the sake of completeness.

LEMMA 2.6. – Let β a standard Brownian motion and λ0. Then, the process X= {X(t)=0^texp(−λ(t−s)) dβ(s),0t1}has the following Karhunen–Loève expansion:

X(t)=^∞

k=1

1 λ²+x_k²

Ykgk(t)

where, for eachk1,xk is the unique positive solution of the equation tan(x)= −¹_λxin the interval[(2k−1)^π₂, (2k+1)^π₂),{gk(t)=Aksin(xkt), k1}is an orthonormal basis ofL²([0,1])with the normalizing constantsAksatisfying sup_k|Ak|2, and{Yk, k1} is a family of orthogonal Gaussian random variables with mean 0 and variance 1, defined byYk=λ²+x_k²₀¹X(t) gk(t) dt, for allk1.

Proof. – Since the caseλ=0 is well known, we will assume thatλ >0. Note thatX is a Gaussian process with covariance function

K(t, s)=

t∧s

0

e⁻^λ(t⁻^u)e⁻^λ(s⁻^u)du= 1 2λ

e⁻^λ(t^∨^s⁻^t^∧^s)−e⁻^λ(t⁺^s), (4)

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s, t ∈ [0,1]. To find the eigenvalues and the eigenvectors of the symmetric operator in L(L²([0,1]))associated with the kernelK, we have to solve the equation

1

0

K(t, s)g(t) dt=µg(s), 0s1, (5)

that is, for 0s1, 1

2λ

e⁻^λs s

0

e^λtg(t) dt+e^λs 1

s

e⁻^λtg(t) dt−e⁻^λs 1

0

e⁻^λtg(t) dt

=µg(s).

Differentiating twice, it is easy to check thatgsatisfies

λ²µ−1g(s)=µg(s), 0s1, (6) with initial conditionsg(0)=0 andλg(1)= −g(1). Observe that Eq. (6) clearly implies thatµ=0.

Setαλ,µ=_λ2µ^µ−1. Thenαλ,µis well-defined and strictly negative. Indeed, suppose that λ²µ−1=0. Theng(s)=0, 0s1, and the initial conditions implyg≡0. Finally, suppose thatαλ,µ>0. In this case, the solution of the differential equation (6) is of the form

g(s)=c1exp s

√αλ,µ

+c2exp

− s

√αλ,µ

,

wherec1, c2are real constants. Then, the initial conditionsg(0)=0 andλg(1)= −g(1) yield

tanh 1

√αλ,µ

= −1 λ

√α1λ,µ

and this equation has no solution.

Consequently, we can assume _λ2µ^µ−1<0, and the solution of (6) is of the form g(s)=c1sin

s |αλ,µ|

+c2cos s

|αλ,µ|

. The conditiong(0)=0 impliesc2=0, andλg(1)= −g(1)yields

tan 1

|αλ,µ|

= −1 λ

1

|αλ,µ|.

Set x = |αλ,µ|⁻^1/2. The relation αλ,µ= _λ2µ^µ−1 implies that the eigenvalues of the operator K form a family{µn;n1}, whereµn=_λ2+¹x_n² and xn is the solution of the equation tan(x)= −¹_λxin the interval[(2n−1)^π₂, (2n+1)^π₂); and the orthonormalized

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eigenfunctions are of the form gn(s)=Ansin(xns),n1. An easy computation shows that|An|2 for alln.

The classical Karhunen–Loève Theorem (see e.g. [10]) finishes the proof. ✷ 3. Onsager–Machlup functional

In this section we compute the Onsager–Machlup functional for our equation, following the usual scheme used for both finite and infinite cases: we apply first the Girsanov transform (Section 3.1) in order to reduce our problem to the evaluation of a functional of the stochastic convolution W^A, up to some easily controlled correction terms (Section 3.2). We are then left with the evaluation of the conditional exponential moments of a stochastic integral with respect to the cylindrical Brownian motion, that can be performed explicitely whenF is a linear operator (Section 3.3). The general case forF can be deduced then by Taylor’s expansion (Section 3.4).

3.1. Application of Girsanov’s transform

Fix a function h∈L²([0,1];H ). Letφ^h be the solution of the infinite dimensional equation

dφ^h(t)=Aφ^h(t) dt +Bh(t) dt, t∈ [0,1],

φ^h(0)=x. (7)

We will compute the Onsager–Machlup functional onL²([0,1];H )at points of the form φ^h. The assumptionh∈L²([0,1];H )is required in order to apply Girsanov’s transform.

Assume also the following conditions:

(h1) For anyt∈ [0,1],F (t, X(t))∈Im(B)a.s. and one of the two following relations holds: for someδ >0,

sup

t∈[0,1]EexpδB⁻¹Ft, X(t)²_H<+∞, or

E

exp 1

2 1

0

B⁻¹Ft, X(t)²_Hdt

<+∞, (h2) there exist a positive constantKsuch that

sup

t∈[0,1]

B⁻¹F (t, x)−B⁻¹F (t, y)_HK|x−y|H

and

sup

t∈[0,1]

B⁻¹F (t, x)_H K1+ |x|H

, for anyx, y∈H.

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Then, using (h1) we can apply Girsanov’s transformation (see [4, Theorem 10.14 and Proposition 10.17]) with W (t) =W (t)+0^tB⁻¹F (s, X(s)) ds and we obtain for any ε >0

PX−φ^h₂ε=E

exp 1

0

B⁻¹Fs, W^A(s)+e^sAx, dW (s)

−1 2 1

0

B⁻¹Fs, W^A(s)+e^sAx²_Hds

1_{e^·Ax+W^A−φ^h2ε}

. Note that to simplify the notation we have denoted Wby W. Since h∈L²([0,1];H ) we can apply again Girsanov’s transformation, now withW (t) =W (t)−0^th(s) ds.We get, for anyε >0,

PX−φ^h₂ε=E

exp 1

0

B⁻¹Fs, W^A(s)+φ^h(s), dW (s)

+ 1

0

B⁻¹Fs, W^A(s)+φ^h(s), h(s)ds

−1 2 1

0

B⁻¹Fs, W^A(s)+φ^h(s)²_Hds

− 1

0

h(s), dW (s)−1 2 1

0

h(s)²_Hds

1_{W^A2ε}

. Then

γε

φ^h=P (X−φ^h2ε) P (W^A2ε) can be written as

exp(7)E

exp ₃

i=1

Ti W^A₂ε

, (8)

with

7:= − 1

0

1

2B⁻¹Aφ^h(t)+Ft, φ^h(t)− ˙φ^h(t)²_Hdt, T1:=

1

0

B⁻¹Fs, W^A(s)+φ^h(s), dW (s),

T2:=

1

0

B⁻¹Fs, W^A(s)+φ^h(s)−B⁻¹Fs, φ^h(s), h(s)ds

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−1 2 1

0

B⁻¹Fs, W^A(s)+φ^h(s)²_H −B⁻¹Fs, φ^h(s)²_Hds

and

T3:= − 1

0

h(s), dW (s).

3.2. Study ofW^Aand Wiener integrals

In this subsection we obtain expressions of W^A and Wiener integrals that will be useful to study the terms obtained in the Girsanov expansion. We apply these expressions to reduce the study of the Onsager–Machlup functional to the study ofT1.

Assuming hypothesis (H1) and (H2), by Lemma 2.6 and decomposition (2) we have W^A(t)=^∞

j=1

t 0

βje⁻^α^j^(t⁻^s)dW^j(s)

ej

=^∞

j=1

∞ k=1

µk,jYk,j(gk,j⊗ej)(t), (9) where µk,j =βj/

α_j²+x_k,j² , and xk,j, Yk,j and gk,j are the xk, Yk and gk defined in Lemma 2.6 when λ=αj. Moreover, {Yk,j, k1, j 1} is a family of independent centered random variables with variance 1.

Givenf ∈L²([0,1]) and e∈H, we denote by f ⊗e the function ofL²([0,1];H ) such that(f⊗e)(s)=f (s)e.Note then that{gk,j⊗ej, k1, j1}is an orthonormal basis ofL²([0,1];H )such that for anyj, k1: Cov(W^A, gk,j ⊗ejL²([0,1];H ))=µ²_k,j. Thus,

W^A²₂=^∞

j=1

∞ k=1

µ²_k,jY_k,j² , with^∞_j,k₌₁µ²_k,j <+∞. Indeed, ifj < j0

∞ k=1

µ²_k,j =β_j² ∞ k=1

1

x_k,j² =Cβ_j². On the other hand, whenjj0,

∞ k=1

µ²_k,j =^∞

k=1

β_j²

α²_j+x_k,j² β_j² α_j²

∞ k=1

1 1+^(2k⁻_4α¹⁾2²^π²

j

<β_j² α²_j

∞

0

1 1+_4α^π²2

j

x²dx

=β_j² αj

∞

0

1

1+^π₄²x²dxCβ_j² αj

. (10)

So

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∞ j,k=1

µ²_k,j C _j

0−1

j=1

β_j²

αj+1 +^∞

j=j₀

β_j² αj+1

αj+1 αj

C

1+ 1

αj₀

_∞

j=1

β_j²

αj+1 <∞. Consider also

hk,j(s):= 1 µk,j

1

s

βje⁻^α^j^(t⁻^s)gk,j(t) dt,

j 1, k1. Then, for anyj 1,{hk,j, k1}is an orthonormal basis of L²([0,1]).

Indeed, thehk,j form an orthogonal family since, by (4) and (5), hm,j, hn,jL²[0,1]

= β_j² µm,jµn,j

1

0

1 s

e⁻^α^j^(t⁻^s)gm,j(t) dt 1

s

e⁻^α^j^(u⁻^s)gn,j(u) du

ds

= β_j² µm,jµn,j

1

0

1

0

Kj(t, u)gm,j(t)gn,j(u) dt du

= gm,j, gn,jL²[0,1], (11) for anym, n1, whereKj denotes the covariance function defined at (4) withλ=αj. Thus, in order to prove that {hk,j, k 1} is a basis, it is sufficient to show that, if h∈L²([0,1]) satisfies hk,j, hL²[0,1]=0 for all k1, then h≡0. But this follows easily from the fact that if for allk1,

0= hk,j, hL²[0,1]= 1 µk,j

1

0

1 s

βje⁻^α^j^(t⁻^s)gk,j(t) dt

h(s) ds

= βj

µk,j

gk,j, ϕ^h_L2[0,1],

thenϕ^h≡0 withϕ^h(t)=₀^te⁻^α^j^(t⁻^s)h(s) ds, and of courseh≡0.

Furthermore,

Yk,j=α_j²+x²_k,j 1

0

t 0

e⁻^α^j^(t⁻^s)dW^j(s)

gk,j(t) dt

= 1

0

βj

µk,j

1

s

e⁻^α^j^(t⁻^s)gk,j(t) dt

dW^j(s)

=Ij(hk,j),

where Ij(l) denotes the Wiener integral of l with respect to W^j, that is Ij(l)= ₁

0 l(s) dW^j(s).

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From these considerations, we get the following lemma:

LEMMA 3.1. – Letl∈L²([0,1];H ), and setU=0¹l(s), dW (s). Then

εlim→0Eexp(U )|W^A₂ε=1.

Proof. – Sincel∈L²([0,1];H ), we have l=^∞

j=1

∞ k=1

πk,j(hk,j⊗ej),

withπk,j= l, hk,j⊗ejL²([0,1];H )and^∞_j₌₁^∞_k₌₁π_k,j² <∞.Furthermore, we have U=^∞

j=1

∞ k=1

πk,jIj(hk,j)=^∞

j=1

∞ k=1

πk,jYk,j. Hence, the result follows easily by Lemma 2.4. ✷

Applying Lemma 3.1, sinceh∈L²([0,1])we directly get that

εlim→0Eexp(cT3)|W^A₂ε=1,

for anyc∈R. On the other hand, on the set{W^A2ε}using (h2) it is easy to check that|T2|Cε, and hence

lim sup

ε→0 Eexp(cT2)|W^A₂ε1,

for any c∈R. Thus, using Lemma 2.2, the only point remaining in order to determine limε↓0γε(φ^h)is the study of the termE[exp(T1)| W^A2ε].

3.3. The linear case

In this subsection we discuss the case of a linear functionF that not depend ont. We obtain the Onsager–Machlup functional and we study carefully the particular case where F can be diagonalized in the same basis as the operatorsAandB.

The main theorem of this part states as follows:

THEOREM 3.2. – Assume that (H1) and (H2) are satisfied, h:[0,1] → H is a function such that h∈L²([0,1];H ), φ^h is defined by (7) and F ∈L(H ) is such that P=B⁻¹F is a bounded operator. Denote by P andR the linear operators defined on L²([0,1];H )such that for anyf ∈L²([0,1]),P (f⊗ej)=f⊗P (e j)andR(f⊗ej)= Rj(f )⊗ej with

(Rjf )(s):=

1

s

βje⁻^α^j^(t⁻^s)f (t) dt.

(13)

Then, for anyj, k1,

(R^∗R)(gk,j⊗ej), gk,j ⊗ej

L²([0,1];H )=CovW^A, gk,j ⊗ej

L²([0,1];H )

and

(i) ifSP R^∗≡¹₂(P R^∗+(P R^∗)^∗)is trace class, then

limε↓0γε

φ^h=exp

− 1

0

1

2B⁻¹(A+F )φ^h(t)− ˙φ^h(t)²_Hdt−Tr(SP R^∗)

, (ii) if_j,kP R^∗(gk,j⊗ej), gk,j ⊗ej = +∞(respectively−∞), then

limε↓0γε

φ^h=0

(respectively+∞).

Proof. –

Step 1. Reduction to a stochastic integral involvingW^A andW.

Since P is a bounded operator, condition (h2) is clearly satisfied. On the other hand, since F ∈L(H ) we have that (A+F ) generates a C0-semigroup such that ₁

0 e^{t (A}⁺^{F )}B²H Sds <∞(see e.g. Goldberg [7, Chapter 5.1]). Thus, following the proof of [4, Theorem 10.20] we get

E

exp 1

2 1

0

B⁻¹FX(t)²_Hdt

exp

C 2

1

0

1+e^{t (A}⁺^{F )}²|x|²H

dt

E

exp C

2 1

0

W^A⁺^F(t)²_Hdt

.

<+∞

and (h1) is clearly satisfied.

Hence, it is sufficient to study limε↓0E[exp(T1)W^A2ε]. Since we are in the linear case, we can writeT₁:=T1(a)+T1(b)with

T1(a)= 1

0

B⁻¹FW^A(s), dW (s),

T1(b)= 1

0

B⁻¹Fφ^h(s), dW (s).

SinceB⁻¹F is a bounded operator,B⁻¹F (φ^h)∈L²([0,1];H )andT1(b) can be handled using Lemma 3.1.

The study of the term T1(a) will follow the ideas presented by Mayer-Wolf and Zeitouni [11].

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Step 2. Expression ofT1(a)in terms of Skorohod integrals.

Notice first that for anyj 1, k1,hk,j =_µ¹_k,jRjgk,j.

Then, using the decompositions given in Section 3.2 (see (9)) we have T1(a)=

1

0

PW^A(s), dW (s)

= ^∞

i,j,k=1

µk,j

1

0

Yk,j

gk,j(s)⊗P (e j), ei

dWⁱ(s)

= ^∞

i,j,k=1

µk,jP e j, ei 1

0

Yk,jgk,j(s) dWⁱ(s). (12) It is worth noting that the random variablesYk,j areF1-measurable. Hence, some of the stochastic integrals appearing in (12) are anticipating. When they are of this kind, we have taken them in the Skorohod sense, and we switch from Itô’s integrals to Skorohod’s ones using the fact that they coincide on the setL²_aof square integrable adapted processes (see for instance [12] for an account on Skorohod’s integrals). Moreover, using [12, Eq. (1.45)], observe that whenj=i

1

0

Yk,jgk,j(s) dW^j(s)=Yk,j

∞ m=1

gk,j, hm,jL²[0,1]Ij(hm,j)− hk,j, gk,jL²[0,1],

and whenj=i 1

0

Yk,jgk,j(s) dWⁱ(s)=Yk,j

∞ m=1

gk,j, hm,iL²[0,1]Ii(hm,i).

Using the fact that hk,j =µ⁻_k,j¹Rjgk,j and Yk,j =Ij(hk,j), we can write (12) in the following way:

T1(a)=

(k,j )=(m,i)

µk,j

µm,i

Yk,jYm,iP e j, eigk,j, Rigm,iL²[0,1]

+^∞

j=1

∞ k=1

Y_k,j² −1P e j, ejgk,j, Rjgk,jL²[0,1]. (13)

Step 3. Expression ofT1(a)in terms ofP andR^∗. Define now the operatorT:)²_N2→)²_N2 by

T(k,j ),(m,i)= µk,j

µm,i

P e j, eigk,j, Rigm,iL²[0,1], (k, j ), (m, i)∈N². (14)